Đề thi WoPhO 2012 Bài 2 môn vật lý

Consider a flow of the viscous fluid through a tube with length L0 and radius r0 Figure 4.. Close to the solid boundary molecules are stuck, while in other regions velocity varies, with

Figure 1: Scheme of the oil

produc-tion process

exert a huge pressure on the petroleum sponge, the oil flows to the surface (see Figure 1) In this problem we neglect capillary and gravity effects on the fluid flow

Figure 2: Representation of the porous medium (grains in white and void space in blue)

Figure 3: Cubic stacking of identical spherical grains

(a) One of the most important characteristics of the reservoir is porosity, which is the fraction of the void space in the rock to the total volume:

ϕ = Vvoid

Vgrains+ Vvoid , (1) where V stands for the volume

To understand meaning of this concept, imagine identical balls (grains of sand) that put in a pile as shown in Figure 3

Find the porosity of the system if the number of balls is infinitely large (0.3 points ) (b) A fluid flow between the grains of sand is controlled by viscosity and permeability Consider

a flow of the viscous fluid through a tube with length L0 and radius r0 (Figure 4) Fluid molecules moves along free paths and collide with each other However, this process is not uniform Close to the solid boundary molecules are stuck, while in other regions velocity varies, with a profile similar to the sketch shown in Figure 4, where y is measured from the axis of the tube

Figure 4: Schematic of the viscous fluid flow in the tube

The reason for this effect is the internal friction of the fluid, or viscosity If two adjacent layers of fluid flow with slightly different speeds, the random sidewise intrusion of some faster molecules into the slower stream will tend to speed up the slower stream, whereas intrusion of slower molecules into the faster stream will tend to slow down the faster stream This effect could be quantified with following well-known equation:

Ff r= −µAf rdv

where Ff r is a friction force which occurs between two thin layers of the fluid separated by

a small distance dy, which have differences in velocities dv; Af r is a contact area on which applied the internal friction force; µ is the fluid property called coefficient of viscosity

Find the velocity distribution v(y) in terms of µ, L0, r0, P1 and P2 Assume that a mean free path of molecules is much smaller than the radius of the tube (0.6 points) (c) Under described conditions fluid will flow through the tube with a flow rate:

q = k0

µπr

2 0

P1− P2

L0

Calculate coefficient k0 in Poiseuille equation (0.3 points) (d) A fluid flow through a porous medium is governed by Darcy’s law:

q = dV

dt =

k

µA

(Pin− Pout)

where dVdt is the amount of fluid transferred through the rock in some period of time; A, L are the cross-sectional area and length of the sample shown in Figure 5; Pin− Pout is the pressure drop; k is the permeability and is the property of the rock

(You can easily recognize some similarity with Fouriers Law for heat transfer Using analogies with heat transfer could significantly help in solution of this problem, because approaches are very similar)

Porous medium can be modeled as a system of twisted tubes (Figure 5), with permeability

k = k0ϕ2 Where k0 is permeability of a straight capillary; ϕ2 accounts for nonlinearity of the tubes in a porous medium with porosity ϕ

Figure 5: Diagaram showing definition of Darcy’s law

Estimate permeability of the system described in 1.a, with radiuses of the balls equal to 10−6

(e) Usually, rock properties are not uniform throughout the reservoir However, it is possible to apply an averaging procedure to find an effective permeability kef f This means that the initial system could be replaced with a new model that has the same sizes and fluid flow parameters with the only difference in permeability, which is uniform throughout the new homogeneous sample To examine this issue, we consider a sample consisting of two different rock types as shown in Figure 6 An incompressible fluid flows through that system with a flow rate q and viscosity µ

Figure 6: Composite rock sample

Calculate pressure at the boundary between two different rocks Pb in terms of q, µ, and

(f) Find the effective permeability of the system kef f (0.4 points)

Often the reservoir can be modeled as a cylinder (see Figure 7) For this problem all properties were averaged out as in the previous part, so the reservoir is assumed homogeneous with uniform permeability k Oil can be viewed as an incompressible fluid with viscosity µ Because the rocks above and below the reservoir are impermeable and the height of the cylinder is much less than its radius (h << R), one can conclude that the fluid flows only in the radial direction

Figure 7: Cylindrical reservoir with a vertical well drilled in the center

(a) Find the velocity of the oil vw inside the well with radius rw = 0.1 m, if the flow rate is 30

m3

day Estimate the fluid velocity in the reservoir near the well vr (0.4 points) (b) The calculated fluid velocity in the reservoir is rather small therefore the reservoir pressure can be treated as a constant for several months or even years, especially if the reservoir is connected with an underground source of water Let Pb be the pressure at the outer boundary

of the reservoir and Pwbe the pressure at the bottom of the well In this part assume that both

Pb and Pw are constants (time-independent values), as well as the radial pressure distribution Calculate pressure drop Pb− Pw , which is required to produce oil with flow rate q

(1.0 points) (c) Make a sketch of pressure distribution in the reservoir P (r) as a function of the distance from

In this part the depletion process will be analyzed for the reservoir shown in Figure 8 The well has a horizontal part, therefore, the fluid flow in the reservoir is linear (h << L)

Figure 8: System used for modeling reservoir depletion This time, pressure at the bottom of the well Pw is constant (hydrostatic column of oil) How-ever, the pressure at the boundary Pb(t) is changing with time, as well as the oil production rate

P ≈ Pb+ Pw

(a) Derive an explicit expression for q(t) in terms of k, µ, cr, ϕ and reservoir dimensions, if the

(b) For the system described in this section, determine the time needed to deplete the oil reservoir

(c) Historically, first attempts to predict reservoir performance were made by doing analogies to electric circuits Draw a simple electric circuit, which is analogous to the system described in

(d) The reason why gas reservoirs cannot be modeled with the circuit as in 3.c, is that gas is highly compressible, with a compressibility being strongly dependent on the applied pressure Compressibility is defined as:

c = −1 V

 dV dP



T

where V is initial volume of the examined sample, dV is isothermal volume change, when additional pressure dP is applied

Assuming natural gas as ideal, derive its compressibility cg as funtion of pressure P

(0.3 points)

Most of the world’s largest oil reservoirs have a different structure, which is not a pile of small balls with fluid between, but a very complex system of porous medium and fractures as shown in Figure

9 (left) Fortunately, such reservoirs could be easily modeled with a stack of sugar cubes, as in Figure 9 (right)

In the model supposed that the production from fractured reservoir goes from the matrix to the fracture and therefrom to the well Thus, matrix does not produce directly into the well Such

a simple model gives incredibly good results for oil production forecasts with an equation:

q = σa

3

where q is a flow rate from matrix to fracture, Pm is average pressure at the matrix, Pf is pressure

at fracture, near the boundary of the sugar cube with a side a, and σ is shape factor, related to the dimensions of the sugar cubes

Figure 9: Idealization of a fractured system

Figure 10: Sugar cube from the inside

The goal of this part of the problem is to estimate shape factor σ Consider a cube with the side

a filled with a porous medium with porosity ϕ, permeability k, and compressibility cr Oil flows with a constant rate q symetrically from the center of the cube to its boundaries, where pressure

is equal Pf, which changes with time t Furthermore, if the well producing at constant flow rate then the cell pressure will decline in such a way that

dPm

(a) Calculate pressure distribution inside the cube Pm(x) in terms of Pf, a, µ, k, and, q

(2.2 points) (b) What is the shape factor for the cube with side a ? (0.5 points)